Cannot duplicate selection
You have proportional editing on (the blue icon in the 3d view header) with a large proportional size (a large circle that appears when you try to transform) so when you try to move the duplicate mesh it moves the entire object and thus appears to have no effect, toggle it on/off with O or by going to the menu and choosing Disable.



After that to duplicate and separate, use Shift D, confirm with left click and then press P > Selection.